Documents often contain various types of elements, including one- and two-dimensional (2D) structures. A one-dimensional structure is an organization of elements whose location can be described with only one axis (i.e., a line). For example, text is an example of a typical one-dimensional (linear) structure. A two-dimensional structure has elements organized in terms of a flat surface whose location is described with two orthogonal axes. Examples of two-dimensional structures are mathematical expressions, chemical expressions, music notations, and so forth. Creating and editing 2D structures is commonly performed using a keyboard and/or mouse to select from drop-down menus. For example, a drop-down menu may contain symbols used in the 2D structure, such as mathematical symbols. Following is an example of a 2D mathematical structure:
  x  =                    -        b            ±                                    b            2                    -                      4            ⁢                                                  ⁢            a            ⁢                                                  ⁢            c                                      2      ⁢                          ⁢      a      
Creating and editing 2D structures using the keyboard often involves a steep learning curve. Usually, a user learns a specialized language that can describe structure from a particular domain. For example, TeX is a language for describing mathematical expressions using keyboard input. Note that in this scenario there is no easy way to switch between 2D domains. If an end user wants to write chemical expressions as well as mathematical expressions, he/she has to learn at least two languages. Drop-down menus can be used in conjunction with keyboard input to reduce the amount of learning, but input in this way is still a slow and tedious process.
Varieties of products that offer such combined experiences exist on the market today. For example, Microsoft Word 2007 has a proprietary language for keyboard input of 2D structures, as well as drop-down menu support for less advanced users. Nevertheless, most of these applications are capable of offering a good user experience for editing 2D structures only in very limited scenarios, such as when linear parts (one-dimensional) of the structure are being changed. As soon as the user wants to change a non-linear expression, or part of an expression, the process becomes very tedious. For more complicated editing operations, whole parts of a typeset 2D structure often have to be erased and typed in from scratch to achieve the desired result.
A tablet PC, or pen computer, is a notebook or slate-shaped mobile computer, equipped with a touch screen or graphics tablet/screen hybrid technology that allows the user to operate the computer with a stylus, digital pen, or fingertip instead of a keyboard or mouse. Tablet PCs offer a more natural form of input, as sketching and handwriting are a much more familiar form of input than a keyboard and mouse, especially for people who are new to computers. Tablet PCs can also be more accessible because those who are physically unable to type can utilize the additional features of a tablet PC to be able to interact with the electronic world.
Natural input applications are available that recognize and store user handwriting on a tablet PC. Natural input applications often store non-text user input as images without any information about the purpose or structure of the image. Thus, this type of input is typically no more useful for editing typeset 2D structures, such as for word processors. The application may also store the user input as a series of ink strokes or as an array of points. Domain-specific natural input applications that support handwriting input for a specified 2D domain (e.g., are equipped with handwriting recognizer for that domain) usually have a very rich correction experience and can therefore provide much more time efficient editing experience, but are often proprietary and not usable with typeset applications or multiple domains.